MODULE: Meaning analysis

TUTORIAL M04: Necessary and sufficient conditions

The concepts of necessary and sufficient conditions help us understand and explain the different kinds of connections between concepts, and how different states of affairs are related to each other.

M04.1 Necessary conditions

To say that X is a necessary condition for Y is to say that it is impossible to have Y without X. In other words, the absence of X guarantees the absence of Y. A necessary condition is sometimes also called "an essential condition". Some examples :

Having four sides is necessary for being a square.
Being brave is a necessary condition for being a good soldier.
Not being divisible by four is essential for being a prime number.

To show that X is not a necessary condition for Y, we simply find a situation where Y is present but X is not. Examples :

Being rich is not necessary for being happy, since a poor person can be happy too.
Being Chinese is not necessary for being a Hong Kong permanent resident, since a non-Chinese can becoming a permanent resident if he or she has lived in Hong Kong for seven years.

Additional remarks about necessary conditions :

We invoke the notion of a necessary condition very often in our daily life, even though we might be using different terms. For example, when we say things like "life requires oxygen", this is equivalent to saying that the presence of oxygen is a necessary condition for the existence of life.
A certain state of affairs might have more than one necessary condition. For example, to be a good concert pianist, having good finger techniques is a necessary condition. But this is not enough. Another necessary condition is being good at interpreting piano pieces.

M04.2 Sufficient conditions

To say that X is a sufficient condition for Y is to say that the presence of X guarantees the presence of Y. In other words, it is impossible to have X without Y. If X is present, then Y must also be present. Again, some examples :

Being a square is sufficient for having four sides.
Being divisible by 4 is sufficient for being an even number.

To show that X is not sufficient for Y, we come up with cases where X is present but Y is not. Examples :

Loving someone is not sufficient for being loved. A person who loves someone might not be loved by anyone perhaps because she is a very nasty person.
Loyalty is not sufficient for honesty because one might have to lie in order to protect the person one is loyal to.

Additional remarks about sufficient conditions :

Expressions such as "If X then Y", or "X is enough for Y", can also be understood as saying that X is a sufficient condition for Y.
Some state of affairs can have more than one sufficient condition. Being blue is sufficient for being colored, but of course being green, being red are also sufficient for being coloured.

M04.3 Four possibilities

Given two conditions X and Y, there are four ways in which they might be related to each other:

X is necessary but not sufficient for Y.
X is sufficient but not necessary for Y.
X is both necessary and sufficient for Y. (or "jointly necessary and sufficient")
X is neither necessary nor sufficient for Y.

This classification is very useful in when we want to clarify how two concepts are related to each other. Here are some examples :

Having four sides is necessary but not sufficient for being a square (since a rectangle has four sides but it is not a square).
Having a son is sufficient but not necessary for being a parent (a parent can have only one daughter).
Being an unmarried man is both necessary and sufficient for being a bachelor.
Being a tall person is neither necessary nor sufficient for being a successful person.

M04.4 Exercises

Q1 Rewrite these claims in terms of necessary and / or sufficient conditions :

(a) You must pay if you want to enter.[Show answer]
Payment is necessary for entrance

(b) A cloud chamber is needed to observe subatomic particles. [Show answer]
A cloud chamber is necessary for observing subatomic particles.

(c) If something is an electron it is a charged particle. [Show answer]
Being an electron is sufficient for being a charged particle.

(d) I will pay for lunch if and only if you pay for dinner. [Show answer]
My paying for lunch is necessary and sufficient for your paying for dinner.

Q2 Suppose Tom is a tall but unsuccessful person. Does it show that (a) being tall is not sufficient for being successful, or (b) being tall is not necessary for being successful? [Show answer]

Yes for (a) and no for (b).

Q3 Discuss how these conditions are related to each other and explain your answers :

(a) not being poor, being rich [Show answer]
The first is necessary but not sufficient for the second, since it is possible to be neither poor nor rich.

(b) being an even number, being divisible by 2 [Show answer]
The first is both necessary and sufficient for the second.

(c) being an intelligent student, being the most intelligent student [Show answer]
Neither necessary nor sufficient. Why?

(d) having ten dollars, having more than five dollars [Show answer]
The first is sufficient but not necessary for the second.

(e) the presence of the rule of law, being a just society [Show answer]
This is a difficult one!

(f) giving money to another person in exchange for a favour, corruption

(g) taking place on a weekday, not being held on Saturday

M04.5 Different kinds of possibility

The concepts of necessary and sufficient conditions relate to the concept of possibility. To say that X is necessary for Y is to say that it is not possible for Y to occur without X. To say that X is sufficient for Y is to say that it is not possible for X to occur without Y. There are, however, different senses of "possibility", and corresponding to these different meanings there are different kinds of necessary and sufficient conditions.

Consider these statements :

It is impossible to be a tall man without being tall.
It is impossible to dissolve gold in pure water.
It is impossible to travel from Hong Kong to New York in less than ten minutes.
It is impossible to visit the army barracks without a permit.

The word "impossible" has different meanings in each of these statements. In the first statement, what is being referred to is logical impossibility. Something is logically impossible if it is contradictory, or against the laws of logic. Thus a round square is a logical impossibility, and it is logically impossible to be a tall man without being tall.

But it is not logically impossible to dissolve gold in water. The laws of logic do not tell us that this cannot happen. Rather, the impossibility is due to the laws of physics and chemistry which happen to hold in our universe. If our universe had contained different laws, then perhaps it is possible to dissolve gold in water. Dissolving gold in water is not logically impossible but empirically impossible. Sometimes this is also known as causal or nomologically impossibility.

The sense in which the third statement is true is again different. The laws of physics probably do not prohibit us from travelling from Hong Kong to New York under ten minutes. What is true is that we have no means to achieve this using current technology. Such a trip is therefore technologically impossible, even though it is both logically and empirically possible. Of course, if the technological obstacles can be overcome then such a trip will then become possible.

Finally, visiting the army barracks without a permit is logically, empirically and technologically possible. After all, one might be able to dig a tunnel to enter the barracks without permission. The sense in which entering without a permit is impossible is in the legal sense. What is meant is that it is illegal or against the relevant regulations to enter the barracks without a permit. Here we are talking about legal impossibility.

M04.6 Different types of necessity and sufficiency

Corresponding to these different notions of possiblity we have different concepts of necessary and sufficient conditions. For example :

Having four sides is logically necessary for being a square.
Being a father is logically sufficient for being a parent.
The presence of oxygen is causally necessary for the proper functioning of the brain.
Passing current through a resistor is causally sufficient for the generation of heat.
Being an adult of over 18 years old is legally sufficient for having the right to vote.
The presence of a witness is legally necessary for a valid marriage.

Note that there might be other types of necessity and possibility as well. For example, a father might advice his son that he must treat his sister well. The sense of "must" here is not legal necessity, as the law does not require us to treat our siblings well. Rather, the sense of necessity might have to do with moral conduct. On the other hand, if the son is told that he must treat his boss well, what is meant might be that being nice to one's boss is required not so much by morality but by rules of prudence.

M04.7 Exercises

Classifying different types of necessary conditions

Click the switches in the diagram below to determine how you might switch on the light.

Now determine the truth or falsity of these claims:

Switch #1 being up is a causally necessary condition for the light to be on. [Show answer]
False
Switch #3 being down is a sufficient condition for the light to be on. [Show answer]
False
All switches being up is a causally sufficient condition for the light to be off. [Show answer]
True
All switches being up is causally necessary for the light to be off. [Show answer]
False
Switch #2 being up and switch #1 being down together is a causally sufficient condition for the light to be on. [Show answer]
Yes, if we assume that the battery and the light are in working condition, and that no part of the circuit is broken. Otherwise not.